Newform orbit 315.2.bb.b

• Label: 315.2.bb.b
• Level: $315$
• Weight: $2$
• Character orbit: 315.bb
• Analytic conductor: $2.515$
• Analytic rank: $0$
• Dimension: $24$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.51528766367\)
Analytic rank:	\(0\)
Dimension:	\(24\)
Relative dimension:	\(12\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 24 q^{4} - 12 q^{10} - 36 q^{19} + 12 q^{25} - 60 q^{31} + 96 q^{40} - 24 q^{46} + 36 q^{49} + 48 q^{61} + 48 q^{64} - 48 q^{70} - 60 q^{79} - 72 q^{85} + 60 q^{91} + 48 q^{94}+O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
89.1	−1.28469	+	2.22514i		0		−2.30084	−	3.98517i	1.33613	+	1.79297i		0		2.64140	−	0.151755i	6.68468				0		−5.70613	+	0.669669i
89.2	−1.28469	+	2.22514i		0		−2.30084	−	3.98517i	2.22083	+	0.260635i		0		−2.64140	+	0.151755i	6.68468				0		−3.43302	+	4.60682i
89.3	−0.956572	+	1.65683i		0		−0.830062	−	1.43771i	−2.17319	−	0.526555i		0		−1.11878	−	2.39757i	−0.650234				0		2.95122	−	3.09692i
89.4	−0.956572	+	1.65683i		0		−0.830062	−	1.43771i	−1.54260	−	1.61876i		0		1.11878	+	2.39757i	−0.650234				0		4.15762	−	1.00738i
89.5	−0.659204	+	1.14177i		0		0.130901	+	0.226727i	−0.729935	+	2.11357i		0		−2.12635	−	1.57437i	−2.98198				0		−1.93205	−	2.22670i
89.6	−0.659204	+	1.14177i		0		0.130901	+	0.226727i	1.46544	−	1.68893i		0		2.12635	+	1.57437i	−2.98198				0		0.962352	+	2.78655i
89.7	0.659204	−	1.14177i		0		0.130901	+	0.226727i	−1.46544	+	1.68893i		0		2.12635	+	1.57437i	2.98198				0		0.962352	+	2.78655i
89.8	0.659204	−	1.14177i		0		0.130901	+	0.226727i	0.729935	−	2.11357i		0		−2.12635	−	1.57437i	2.98198				0		−1.93205	−	2.22670i
89.9	0.956572	−	1.65683i		0		−0.830062	−	1.43771i	1.54260	+	1.61876i		0		1.11878	+	2.39757i	0.650234				0		4.15762	−	1.00738i
89.10	0.956572	−	1.65683i		0		−0.830062	−	1.43771i	2.17319	+	0.526555i		0		−1.11878	−	2.39757i	0.650234				0		2.95122	−	3.09692i
89.11	1.28469	−	2.22514i		0		−2.30084	−	3.98517i	−2.22083	−	0.260635i		0		−2.64140	+	0.151755i	−6.68468				0		−3.43302	+	4.60682i
89.12	1.28469	−	2.22514i		0		−2.30084	−	3.98517i	−1.33613	−	1.79297i		0		2.64140	−	0.151755i	−6.68468				0		−5.70613	+	0.669669i
269.1	−1.28469	−	2.22514i		0		−2.30084	+	3.98517i	1.33613	−	1.79297i		0		2.64140	+	0.151755i	6.68468				0		−5.70613	−	0.669669i
269.2	−1.28469	−	2.22514i		0		−2.30084	+	3.98517i	2.22083	−	0.260635i		0		−2.64140	−	0.151755i	6.68468				0		−3.43302	−	4.60682i
269.3	−0.956572	−	1.65683i		0		−0.830062	+	1.43771i	−2.17319	+	0.526555i		0		−1.11878	+	2.39757i	−0.650234				0		2.95122	+	3.09692i
269.4	−0.956572	−	1.65683i		0		−0.830062	+	1.43771i	−1.54260	+	1.61876i		0		1.11878	−	2.39757i	−0.650234				0		4.15762	+	1.00738i
269.5	−0.659204	−	1.14177i		0		0.130901	−	0.226727i	−0.729935	−	2.11357i		0		−2.12635	+	1.57437i	−2.98198				0		−1.93205	+	2.22670i
269.6	−0.659204	−	1.14177i		0		0.130901	−	0.226727i	1.46544	+	1.68893i		0		2.12635	−	1.57437i	−2.98198				0		0.962352	−	2.78655i
269.7	0.659204	+	1.14177i		0		0.130901	−	0.226727i	−1.46544	−	1.68893i		0		2.12635	−	1.57437i	2.98198				0		0.962352	−	2.78655i
269.8	0.659204	+	1.14177i		0		0.130901	−	0.226727i	0.729935	+	2.11357i		0		−2.12635	+	1.57437i	2.98198				0		−1.93205	+	2.22670i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
7.d	odd	6	1	inner
15.d	odd	2	1	inner
21.g	even	6	1	inner
35.i	odd	6	1	inner
105.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	315.2.bb.b	✓	24
3.b	odd	2	1	inner	315.2.bb.b	✓	24
5.b	even	2	1	inner	315.2.bb.b	✓	24
5.c	odd	4	2		1575.2.bk.i		24
7.c	even	3	1		2205.2.g.b		24
7.d	odd	6	1	inner	315.2.bb.b	✓	24
7.d	odd	6	1		2205.2.g.b		24
15.d	odd	2	1	inner	315.2.bb.b	✓	24
15.e	even	4	2		1575.2.bk.i		24
21.g	even	6	1	inner	315.2.bb.b	✓	24
21.g	even	6	1		2205.2.g.b		24
21.h	odd	6	1		2205.2.g.b		24
35.i	odd	6	1	inner	315.2.bb.b	✓	24
35.i	odd	6	1		2205.2.g.b		24
35.j	even	6	1		2205.2.g.b		24
35.k	even	12	2		1575.2.bk.i		24
105.o	odd	6	1		2205.2.g.b		24
105.p	even	6	1	inner	315.2.bb.b	✓	24
105.p	even	6	1		2205.2.g.b		24
105.w	odd	12	2		1575.2.bk.i		24

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
315.2.bb.b	✓	24	1.a	even	1	1	trivial
315.2.bb.b	✓	24	3.b	odd	2	1	inner
315.2.bb.b	✓	24	5.b	even	2	1	inner
315.2.bb.b	✓	24	7.d	odd	6	1	inner
315.2.bb.b	✓	24	15.d	odd	2	1	inner
315.2.bb.b	✓	24	21.g	even	6	1	inner
315.2.bb.b	✓	24	35.i	odd	6	1	inner
315.2.bb.b	✓	24	105.p	even	6	1	inner
1575.2.bk.i		24	5.c	odd	4	2
1575.2.bk.i		24	15.e	even	4	2
1575.2.bk.i		24	35.k	even	12	2
1575.2.bk.i		24	105.w	odd	12	2
2205.2.g.b		24	7.c	even	3	1
2205.2.g.b		24	7.d	odd	6	1
2205.2.g.b		24	21.g	even	6	1
2205.2.g.b		24	21.h	odd	6	1
2205.2.g.b		24	35.i	odd	6	1
2205.2.g.b		24	35.j	even	6	1
2205.2.g.b		24	105.o	odd	6	1
2205.2.g.b		24	105.p	even	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 12T_{2}^{10} + 102T_{2}^{8} + 420T_{2}^{6} + 1260T_{2}^{4} + 1764T_{2}^{2} + 1764 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\).